I believe that mathematics should be taught, not collaboratively explored; algebra and geometry are better than a vague course of Integrated Math; spiraling doesn't work nearly as well as learning it properly the first time; "I don't DO math" should be an incentive rather than an excuse. "I don't DO English" should be treated the same way.

Sunday, April 20, 2014

There are a couple of things we really need you to stop saying. The first:

"The right answer isn't important. It's knowing what you're doing."

No matter how you parse this, it's ridiculous. The right answer is the whole point of doing the problem ... has always been, is now, and will always be. The "knowing what you are doing part" leads to the right answer. If it doesn't, then you don't know what you are doing.

Variations on this include: "It's the concept that matters" and "We're putting the emphasis on the method." It shows up as sarcastic responses from Institute Professionals and college professors, too:

@MathCurmudgeon Good thing math is only about answer-getting, then! Phew!
— xxxx (@xxxx)

What we should be saying is "The right answer is vitally important ... so important that we also want students to explain the method and how we all know the answer is correct; they must be able to detect an error if it occurs and describe how to fix it so that the solution IS correct."

If you go to the trouble of having the students communicate, verbally or in writing, how they solved a problem, then you are focused on the right answer ... there would be no need to explain anything, or fix errors, if you didn't care about it. You'd take any randomly achieved answer as long as it was correct, and move on.

Just cancel the 6s.

Does anyone ... ANYONE ... seriously think that the right answer doesn't
matter here? I don't consider this a "right answer" even though it
looks like it.

I'm going with "No."

You'd take a wrong answer that looked like a right answer if you weren't paying attention.

Just cancel the x² from numerator and denominator.

One hallmark of mathematical understanding is the ability to justify, in a way appropriate to the student’s mathematical maturity, why a particular mathematical statement is true or where a mathematical rule comes from. There is a world of difference between a student who can summon a mnemonic device to expand a product such as (a + b)(x + y) and a student who can explain where the mnemonic comes from.

That is a far cry from " The right answer isn't important."

For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge.

You can't detect errors unless you know the right answer, or at least have a sense of what that right answer should be.

Even the infamous "Letter to Jack" assumed that the kid could get the right answer, then could find the error made by the other kid ... the assignment took this two steps beyond the right answer: explain the error to the other kid and help him fix it.

I often "let them in on a secret" and share the mnemonics after they get the understanding ... especially if the mnemonics speed up computation so we can get on with what we are actually doing, but every teacher worth his salt knows that you have to periodically make sure that random, blind luck isn't at play.

The last reason that we need to stop saying "The right answer isn't as important as knowing what you're doing" is that too often we teachers are speaking to people who don't know that it is merely step 1, namely school boards, administrators and parents.

I watched a young teacher from another school give a presentation to her school board. Among other weird things, she came out with this statement ... immediately, board members latched onto it.

"What do you mean? Of course the right answer matters."
"I've been in business for forty years; every time, the right answer matters."

She doubled down ... "No, they need to know HOW they are solving the problem." No one was buying it, nor should they have. Whether she didn't understand their concerns herself or honestly didn't believe that the right answer was so vital, she certainly couldn't communicate her stance to the Board.

A blind acceptance and repetition of poorly-understood Twitter broadsides and mindless slogans is the rhetorical equivalent of canceling the sixes.

That makes us all look bad and we're gonna need you to stop saying it.

5 comments:

I'll agree that to say that the right answer is not important at all sends the wrong message, however I have to disagree with quite a bit of the rest of your post. First, the school board members who say "I've been in business 40 years; every time the right answer matters" are looking at the world through rose colored glasses. Very rarely in business, or really any field, is there a cut and dry "right answer." Answers seem right in hindsight because they work out for the better, and answers seem wrong in hindsight because they don't turn out so great. In the moment, however, those answers are grey and uncertain and what matters is the manner in which a person arrived at them.

Beyond that argument, however, I still take issue with your claim that the right answer is step #1. Sure if all we are doing is relatively basic, procedural problems like the ones you posted images of, then the right answer is important. But only because there really isn't much else in the problem that could be more important. The problem is that even if we encourage students to justify their answers or force them explain why rules they applied work, they're not explaining them because they need or want to, but because it's part of the game. We know the answer, and they know we know the answer. This song and dance doesn't encourage them to engage fully with the material. In fact it often encourages them to engage just enough to satisfy their teacher that they didn't get lucky. Or in some causes, the students think "this game is silly, and I'm not even going to play" and then they don't engage at all.

The issue, at least as I see it, is that the problems that you use as your examples are just exercises (and these problems constitute the vast majority of what students see in K-12 math). If we don't ever give students a chance to engage in real problems--problems where there multiple right answers or multiple productive approaches--then we are hiding mathematics from them. Expanding (a+b)(x+y) is not math. Explaining why (a+b)(x+y) is also not math. Mathematicians do not publish papers on how well they can repeat back a result that their teacher explained to them once. Certainly, many fields (physics, business, etc.) make great use of the results of math, but we cannot confuse an ability to fluently calculate using procedures in applied situations with learning math.

Learning these procedures, learning to get the right answer, and learning to know when you didn't get the right answer are admittedly important parts of being able to be successful in mathematics. But when it comes to learning what mathematics really is and learning to truly think mathematically, it's not that the right answer isn't important, it's just that it's irrelevant.

"But when it comes to learning what mathematics really is and learning to truly think mathematically, it's not that the right answer isn't important, it's just that it's irrelevant."

And you assume that any more than a small fraction of the students want to focus on math, want to "truly think mathematically" and learn "What mathematics really is"? It's attitudes like that that make your opinion irrelevant to them.

Some day, some of them will get there, but they can't just jump directly to your city in the clouds; being so ethereal and holier-than-thou isn't helping to convince them to want to bother.

I'm not talking about making all kids into mathematicians or "city in the clouds" stuff. We need to think very carefully about what we are teaching kids. 90% of what is in the curriculum can be done faster, more efficiently, and more accurately on the smartphone in their pockets. We don't need to teach them mathematical thinking so they can become mathematicians, we need to teach them this so they can use math to reason about statistics they see in the paper or make informed decisions about buying a house. The fact that you say "some day, some of them will get there" implies that you think many of them won't get there. What about these kids? Why are you teaching them? What do you hope they get out of your class? I'm not arguing for "holier-than-thou" math that isn't interesting to them. In fact, I would say that simplifying rational polynomial expressions because they need to do them to solve "all important" limits in calculus is exactly the holier-than-thou math that disinterests them in the first place. Students don't have to try to tackle the Riemann Hypothesis to experience real mathematics. Take the border problem (http://www2.edc.org/cme/showcase/KY/TheBorderProblem.pdf) or this video (https://www.youtube.com/watch?v=Ien-86bXCrI) both of which are by Jo Boaler. These problems are both easily accessible by nearly all students, yet they allow students to explore the concepts deeply. And in both these problems, the right answer is 1) not unique and 2) much less important than the process used to get the answer. If the average person is going to use any of the math we teach them outside of class, it is going to be thinking skills, pattern recognition skills, number sense, or model building. Both experience and research show that they are not going to use the quadratic formula--even if we make them justify it. When was the last time, outside of a math class or an academic setting, that you expanded the product of two binomials?

Aran, I generally agree with Curmudgeon, but you have made me think. Yet, I don't think students can get really comfortable with the (very valuable) skills needed to understand statistics or make good choices when buying a house, UNLESS they have some personal ownership of the basic arithmetic and algebra operations, ownership that in my experience young people don't develop unless they can do the operations without a calculator. That's not to say that we shouldn't use calculators once we have that ownership (although I'm careful to keep my hand in with a certain amount of in-my-head and paper/pencil practice).

And then there are the students who are going to use serious math later on, and we don't know which students those will turn out to be when they are 14 (though we can make pretty accurate guesses sometimes). Those students may not be mathematicians per se, so much as engineers, accountants, meteorologists, construction managers, radiologists, epidemiologists, supply chain managers, and so forth. They are not necessarily thrilled by the beauty of abstract math, but they have to be very skilled at using mathematical thinking to analyze problems and choose methods for solving them; they are highly oriented towards getting the right answer at the same time.

Thanks for taking my comment out of context and labeling me as an Institute Professional. I spent many years teaching high school and now work with teachers and schools around the country as a curriculum writer and a PD instructor.

The curriculum I am a lead writer on, CME Project, has this as its fundamental organizing principle:

The widespread utility and effectiveness of mathematics come not just from mastering specific skills, topics, and techniques, but more importantly, from developing the ways of thinking —the habits of mind—used to create the results.

You're wrong if you feel I don't think accuracy and correctness are important. What stinks is the weird single-purpose tools kids sometimes are asked to memorize. These tools produce correct answers without understanding, and that is the context from which you pulled my Twitter comment. If correct answers were all that mattered, why bother teaching for understanding if there a 100% guaranteed correct method?

I guess I'll be more careful in future conversation, and I am not yet old enough to have earned the capital letters as a full Institute Professional.

About Me

I'm a high-school math teacher completely frustrated with new math, reform math, fuzzy math, the color of math, talking about math, literacy across the curriculum and all those other things that get in the way of students actually learning math, not to mention the ever-present "You need to help raise our scores by taking one day a week to go over test-taking skills" and other administrative folderol.